Newform orbit 1700.2.c.a

• Label: 1700.2.c.a
• Level: $1700$
• Weight: $2$
• Character orbit: 1700.c
• Analytic conductor: $13.575$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1700.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5745683436\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 68)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{3} - 3 \beta q^{7} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times\).

\(n\)	\(477\)	\(851\)	\(1601\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

101.1

	−	1.41421i
		1.41421i

0

−

1.41421i

0

0

0

4.24264i

0

1.00000

0

101.2

0

1.41421i

0

0

0

−

4.24264i

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1700.2.c.a		2
5.b	even	2	1		68.2.b.a	✓	2
5.c	odd	4	2		1700.2.g.a		4
15.d	odd	2	1		612.2.b.a		2
17.b	even	2	1	inner	1700.2.c.a		2
20.d	odd	2	1		272.2.b.c		2
35.c	odd	2	1		3332.2.b.a		2
40.e	odd	2	1		1088.2.b.f		2
40.f	even	2	1		1088.2.b.e		2
60.h	even	2	1		2448.2.c.d		2
85.c	even	2	1		68.2.b.a	✓	2
85.g	odd	4	2		1700.2.g.a		4
85.j	even	4	2		1156.2.a.c		2
85.m	even	8	2		1156.2.e.a		2
85.m	even	8	2		1156.2.e.b		2
85.p	odd	16	8		1156.2.h.d		8
255.h	odd	2	1		612.2.b.a		2
340.d	odd	2	1		272.2.b.c		2
340.n	odd	4	2		4624.2.a.n		2
595.b	odd	2	1		3332.2.b.a		2
680.h	even	2	1		1088.2.b.e		2
680.k	odd	2	1		1088.2.b.f		2
1020.b	even	2	1		2448.2.c.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.2.b.a	✓	2	5.b	even	2	1
68.2.b.a	✓	2	85.c	even	2	1
272.2.b.c		2	20.d	odd	2	1
272.2.b.c		2	340.d	odd	2	1
612.2.b.a		2	15.d	odd	2	1
612.2.b.a		2	255.h	odd	2	1
1088.2.b.e		2	40.f	even	2	1
1088.2.b.e		2	680.h	even	2	1
1088.2.b.f		2	40.e	odd	2	1
1088.2.b.f		2	680.k	odd	2	1
1156.2.a.c		2	85.j	even	4	2
1156.2.e.a		2	85.m	even	8	2
1156.2.e.b		2	85.m	even	8	2
1156.2.h.d		8	85.p	odd	16	8
1700.2.c.a		2	1.a	even	1	1	trivial
1700.2.c.a		2	17.b	even	2	1	inner
1700.2.g.a		4	5.c	odd	4	2
1700.2.g.a		4	85.g	odd	4	2
2448.2.c.d		2	60.h	even	2	1
2448.2.c.d		2	1020.b	even	2	1
3332.2.b.a		2	35.c	odd	2	1
3332.2.b.a		2	595.b	odd	2	1
4624.2.a.n		2	340.n	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1700, [\chi])\):

\( T_{3}^{2} + 2 \)

\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 2 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 18 \)
$11$	\( T^{2} + 2 \)